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Car-Like Robot: How to Park a Car? (Nonholonomic Planning). Types of Path Constraints. Local constraints: e.g., lie in free space Differential constraints: e.g., have bounded curvature Global constraints: e.g., have minimal length. Types of Path Constraints.

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types of path constraints
Types of Path Constraints
  • Local constraints: e.g., lie in free space
  • Differential constraints: e.g., have bounded curvature
  • Global constraints: e.g., have minimal length
types of path constraints1
Types of Path Constraints
  • Local constraints: e.g., lie in free space
  • Differential constraints: e.g., have bounded curvature
  • Global constraints: e.g., have minimal length
car like robot

f

q

f

Car-Like Robot

L

q

y

x

Configuration space is 3-dimensional: q = (x, y,q)

example car like robot

f

dx/dt = v cosq

dy/dt = v sinq

dx sinq – dy cosq = 0

q

dq/dt = (v/L) tan f

f

|f| <F

Example: Car-Like Robot

L

q

y

x

Configuration space is 3-dimensional: q = (x, y,q)

But control space is 2-dimensional: (v, f) with |v| = sqrt[(dx/dt)2+(dy/dt)2]

example car like robot1

f

dx/dt = v cosq

dy/dt = v sinq

dx sinq – dy cosq = 0

q

dq/dt = (v/L) tan f

f

|f| <F

Example: Car-Like Robot

L

q

y

x

Lower-bounded turning radius

how can this work tangent space velocity space

f

q

(x,y,q)

L

(dx,dy,dq)

q

q

f

y

x

y

dx/dt = v cosq

dy/dt = v sinq

(dx,dy)

x

q

dq/dt = (v/L) tan f

|f|<F

How Can This Work?Tangent Space/Velocity Space

dx sinq – dy cosq = 0

how can this work tangent space velocity space1

f

q

(x,y,q)

L

(dx,dy,dq)

q

q

f

y

x

y

dx/dt = v cosq

dy/dt = v sinq

(dx,dy)

x

q

dq/dt = (v/L) tan f

|f|<F

How Can This Work?Tangent Space/Velocity Space
type 1 maneuver

CYL(x,y,dq,h)

h

h

dq

(x,y,q0)

  • = 2rtandq

d = 2r(1/cosdq - 1) > 0

(x,y)

Type 1 Maneuver

q

r

dq

dq

 Allows sidewise motion

r

When dq 0, so does d and the cylinder becomes arbitrarily small

type 2 maneuver
Type 2 Maneuver

 Allows pure rotation

coverage of a path by cylinders

+

q

q’

Coverage of a Path by Cylinders

q

y

x

Path created ignoring the car constraints

drawbacks
Drawbacks
  • Final path can be far from optimal
  • Not applicable to car that can only move forward (e.g., think of an airplane)
reeds and shepp paths1
Reeds and Shepp Paths

CC|C0

CC|C

C|CS0C|C

Given any two configurations,the shortest RS paths betweenthem is also the shortest path

example of generated path
Example of Generated Path

Holonomic

Nonholonomic

other technique control based sampling

dx sinq – dy cosq = 0

dx/dt = v cos q

dy/dt = v sin q

dq/dt = (v/L) tan f

|f| <F

Other Technique: Control-Based Sampling
  • 1. Select a node m
  • 2. Pick v, f, and dt
  • 3. Integrate motion from m
  • new configuration
other technique control based sampling1
Other Technique: Control-Based Sampling

Indexing array:A 3-D grid is placed over the configuration space. Each milestone falls into one cell of the grid. A maximum number of milestones is allowed in each cell (e.g., 2 or 3).

Asymptotic completeness:If a path exists, the planner is guaranteed to find one if the resolution of the grid is fine enough.

computed paths
Computed Paths

Tractor-trailer

Car That Can Only Turn Left

jmax=45o, jmin=22.5o

jmax=45o

other similar robots moving objects nonholonomic
Other “Similar” Robots/Moving Objects (Nonholonomic)
  • Rolling-with-no-sliding contact (friction), e.g.: car, bicycle, roller skate
  • Submarine, airplane
  • Conservation of angular momentum: satellite robot, under-actuated robot, catWhy is it useful?

- Fewer actuators: design simplicity, less weight

- Convenience (think about driving a car with 3 controls!)

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